Question

How do you divide `(4+2i )/( 1-i)`?

Answer 1

`1+3i`

Explanation:

You must eliminate the complex number in the denominator by multiplying by its conjugate:

`(4+2i)/(1-i)=((4+2i)(1+i))/((1-i)(1+i))`

`(4+4i +2i+2i^2)/(1-i^2)`

`(4+6i-2)/(1+1)`

`(2+6i)/2`

`1+3i`

Answer 2

1 + 3i

Explanation:

Require the denominator to be real . To achieve this multiply the numerator and the denominator by the complex conjugate of the denominator.

If (a + bi ) is a complex number then (a - bi) is the conjugate

here the conjugate of (1 - i ) is (1 + i )

now `((4 + 2i )(1 + i ))/((1 - i)(1 + i ))`

distribute the brackets to obtain :

`(4 + 6i + 2i^2 )/(1 - i^2 )`

note that `i^2 = (sqrt(-1)^2 )= - 1`

hence `(4 + 6i - 2 )/(1 + 1 ) = (2 + 6i )/2 = 2/2 + (6i)/2 = 1 + 3i`